Polar Coordinates and Complex Numbers

A polar point is written $(r,\theta)$, where $r$ is the directed distance from the origin (the pole) and $\theta$ is the angle from the positive $x$-axis (the polar axis).

• If $r>0$, move $r$ units along the terminal side of $\theta$.
• If $r<0$, move $|r|$ units in the opposite direction (add $180^\circ$).

Every polar point has infinitely many representations:

The pole itself is $(0,\theta)$ for any angle $\theta$.

With the pole at the origin and polar axis along the positive $x$-axis:

When finding $\theta$, always check the quadrant of $(x,y)$.

Swap between forms with the same relationships, plus these handy substitutions:

Polar $\to$ rectangular: multiply by $r$ or use identities to remove $r$ and $\theta$. Rectangular $\to$ polar: replace $x,y$ and simplify; solve for $r$ if possible.

• Circles: $r=a$ (centered at pole); $r=a\cos\theta$ or $r=a\sin\theta$ (through the pole, diameter $|a|$).
• Cardioids: $r = a \pm a\cos\theta$ or $r = a\pm a\sin\theta$ — a heart shape with one dimple at the pole.
• Limaçons: $r = a \pm b\cos\theta$ or $r = a\pm b\sin\theta$. Compare $\tfrac{a}{b}$: $\tfrac{a}{b}<1$ inner loop; $\tfrac{a}{b}=1$ cardioid; $1<\tfrac{a}{b}<2$ dimpled; $\tfrac{a}{b}\ge 2$ convex.
• Roses: $r = a\cos(n\theta)$ or $r = a\sin(n\theta)$. Petal count: $n$ petals if $n$ is odd, $2n$ petals if $n$ is even.

A complex number $z = a + bi$ is plotted as the point $(a,b)$: the real part $a$ on the horizontal axis, the imaginary part $b$ on the vertical axis. Its

The shorthand $r(\cos\theta + i\sin\theta)$ is often written $r\,\text{cis}\,\theta$.

For $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$:

Multiply the moduli, add the angles (divide moduli, subtract angles).

For $z = r(\cos\theta + i\sin\theta)$ and any positive integer $n$:

Raise the modulus to the power; multiply the angle by $n$.

Every nonzero $z = r(\cos\theta + i\sin\theta)$ has exactly $n$ complex $n$th roots:

All roots share modulus $\sqrt[n]{r}$ and are spaced $\dfrac{360^\circ}{n}$ apart around a circle.

The $n$th roots of unity are the $n$ solutions of $z^n = 1$. Writing $\omega = \cos\!\frac{360^\circ}{n} + i\sin\!\frac{360^\circ}{n}$ (the primitive root), every root is a power of $\omega$:

They sit at the vertices of a regular $n$-gon on the unit circle, starting at $\omega^0 = 1$. Two beautiful facts (for $n\ge 2$):

If $z = \cos\theta + i\sin\theta$ lies on the unit circle, then $\tfrac1z = \bar z = \cos\theta - i\sin\theta$. By DeMoivre,

so adding and subtracting gives the two workhorse identities

These turn powers of $\cos\theta$ and $\sin\theta$ into sums of $\cos n\theta$ and $\sin n\theta$ (and back).

The set of points $z$ whose distance to $a$ is a fixed multiple $k>0$ of its distance to $b$ is:

• the perpendicular bisector of $a$ and $b$ when $k=1$ (equal distances);
• an Apollonius circle when $k\ne 1$ — squaring $|z-a|^2 = k^2|z-b|^2$ and expanding $z=x+iy$ produces $x^2+y^2+(\cdots)x+(\cdots)y+(\cdots)=0$, the equation of a circle.

The circle is symmetric about the line through $a$ and $b$ and shrinks toward the nearer point as $k\to0$ or $k\to\infty$.

The region swept by the ray to a polar curve $r = f(\theta)$ as $\theta$ runs from $\alpha$ to $\beta$ has area

This comes from summing thin circular sectors of angle $d\theta$ and radius $r$, each of area $\tfrac12 r^2\,d\theta$. Setup cautions: choose $\alpha,\beta$ that trace the region exactly once; for one petal of a rose, integrate between consecutive zeros of $r$; use symmetry to integrate a half and double.

To find where $r = f(\theta)$ and $r = g(\theta)$ meet, solving $f(\theta) = g(\theta)$ is not enough. Because a single point has many polar names, also check:

• the pole separately — it lies on a curve if $r=0$ for some $\theta$, even if the $\theta$ values differ;
• alternate representations, e.g. $f(\theta) = -g(\theta + 180^\circ)$, and shifts by full turns.

Always sketch both curves to confirm the true intersection points.